With temperatures dipping into the single digits at Yongin Mireu Stadium, Anthony Caceres picked out Klimala with a ball over the Jeonbuk defence to set him up for his first goal in the 36th minute, before the Polish striker provided a two-goal cushion from range in the 66th.
Never having won a knockout game in Asian competition until this campaign, the Sky Blues will now return to Australia for next week's second leg at the Sydney Football Stadium in the box seat to book a place in the ACL2's final four, almost certainly against Japanese outfit Sanfrecce Hiroshima.
Progression through to the semis would earn Sydney more than a million dollars in prize money, with further riches up for grabs, as well as a place in next season's ACL Elite, should they advance through to the final.
After Sydney's Wataru Kamijo blazed wide with a volley, and Jeonbuk's Jeon Se-jin forced Harrison Devenish-Meares into a save with a header in the early exchanges, the visitors took advantage of the game stretching as halftime neared to spring Klimala through for the lead.
It took a sharp double-save from Song Bum-keun to deny the Polish striker a second just three minutes later, before Devenish-Meares was forced into action in the 41st minute to save from Jeonbuk talisman Andrea Compagno.
Compagno repeatedly loomed as the hosts pushed to get back into the game.
But their efforts left them exposed to a sucker-punch against the run of play, one delivered when Klimala had enough time to control a pass from Leo Sena and drive an effort into the bottom corner.
